A007272 Super ballot numbers: 60*(2n)!/(n!*(n+3)!).
10, 5, 6, 10, 20, 45, 110, 286, 780, 2210, 6460, 19380, 59432, 185725, 589950, 1900950, 6203100, 20470230, 68234100, 229514700, 778354200, 2659376850, 9148256364, 31667041260, 110248217720, 385868762020, 1357193576760, 4795417304552, 17015996887120, 60619488910365
Offset: 0
References
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Matthew House, Table of n, a(n) for n = 0..1677
- Emily Allen and Irina Gheorghiciuc, A Weighted Interpretation for the Super Catalan Numbers, J. Int. Seq. 17 (2014) # 14.10.7.
- David Callan, A combinatorial interpretation for a super-Catalan recurrence, arXiv:math/0408117 [math.CO], 2004.
- Ira M. Gessel, Super ballot numbers, J. Symbolic Comp., 14 (1992), 179-194
- Ira M. Gessel and Guoce Xin, A Combinatorial Interpretation of the Numbers 6(2n)!/n!(n+2)!, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.3.
- Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 96.
Programs
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Maple
seq(10*(2*n)!/(n!)^2/binomial(n+3,n), n=0..26); # Zerinvary Lajos, Jun 28 2007
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Mathematica
Table[60(2n)!/(n!(n+3)!), {n, 0, 30}] (* Jean-François Alcover, Jun 02 2019 *) -
PARI
a(n)=if(n<0, 0, 60*(2*n)!/n!/(n+3)!) /* Michael Somos, Feb 19 2006 */
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PARI
{a(n)=if(n<0, 0, n*=2; n!*polcoeff( 10*besseli(3,2*x+x*O(x^n)), n))} /* Michael Somos, Feb 19 2006 */ -
Sage
def A007272(n): return -(-4)^(3 + n)*binomial(5/2, 3 + n)/2 print([A007272(n) for n in range(30)]) # Peter Luschny, Nov 04 2021
Formula
G.f.: (11-32*x+9*sqrt(1-4*x))/(1-3*x+(1-x)*sqrt(1-4*x)).
E.g.f.: Sum_{n>=0} a(n)*x^(2n)/(2n)! = 60*BesselI(3, 2x)/x^3.
E.g.f.: (BesselI(0, 2*x)*(2*x+16*x^2)-BesselI(1, 2*x)*(2+6*x+16*x^2))*exp(2*x)/x^2.
Integral representation as the n-th moment of a positive function on [0, 4]: a(n) = Integral_{x=0..4} x^n*(4-x)^(5/2)/(2*Pi*x^(1/2)) dx. This representation is unique. - Karol A. Penson, Dec 04 2001
a(n) = 10*(2*n)!*[x^(2*n)](hypergeometric([],[4],x^2)). - Peter Luschny, Feb 01 2015
(n+3)*a(n) +2*(-2*n+1)*a(n-1)=0. - R. J. Mathar, Mar 06 2018
a(n) = -(-4)^(3+n)*binomial(5/2, 3+n)/2. - Peter Luschny, Nov 04 2021
From Amiram Eldar, Mar 24 2022: (Start)
Sum_{n>=0} 1/a(n) = 4/9 + 28*Pi/(3^5*sqrt(3)).
Sum_{n>=0} (-1)^n/a(n) = 38/1875 - 56*log(phi)/(5^4*sqrt(5)), where phi is the golden ratio (A001622). (End)
From Peter Bala, Mar 11 2023: (Start)
a(n) = Sum_{k = 0..2} (-1)^k*4^(2-k)*binomial(n,k)*Catalan(n+k) = 16*Catalan(n) - 8*Catalan(n+1) + Catalan(n+2), where Catalan(n) = A000108(n). Thus a(n) is an integer for all n.
a(n) is odd if n = 2^k - 3, k >= 2, else a(n) is even. (End)